Averages
What is Averages?
A central value of a set of numbers, calculated by dividing the sum of the numbers by the count of the numbers.
Key formula / rule: Arithmetic Mean (Average)
Key points
- Define and calculate the arithmetic mean for a given set of numbers.
- Calculate the sum of observations given the average and number of observations.
- Solve problems involving the addition or removal of observations and their impact on the average.
- Solve problems involving the replacement of observations and its impact on the average.
Common exam trap
Incorrectly calculating the sum of observations.
Definitions
- Term
Average (Arithmetic Mean)
- Meaning
A central value of a set of numbers, calculated by dividing the sum of the numbers by the count of the numbers.
- Term
Weighted Average
- Meaning
An average that takes into account the relative importance or frequency (weights) of each value in a dataset.
- Term
Observation
- Meaning
A single data point or value within a dataset.
Learning objectives
Define and calculate the arithmetic mean for a given set of numbers.
Calculate the sum of observations given the average and number of observations.
Solve problems involving the addition or removal of observations and their impact on the average.
Solve problems involving the replacement of observations and its impact on the average.
Apply the concept of weighted average to solve real-world problems.
Solve problems involving averages of consecutive numbers or specific series.
Formulae
- Name
Arithmetic Mean (Average)
- Note
Where Σx is the sum of all observations and n is the total number of observations.
- Expression
Average = Σx / n
- Name
Sum of Observations
- Note
Used to find the total sum when average and count are known.
- Expression
Σx = Average × n
- Name
Weighted Average
- Note
Where xi are the observations and wi are their respective weights.
- Expression
Weighted Average = (Σ(xi * wi)) / (Σwi)
- Name
Average of Consecutive Numbers (Arithmetic Progression)
- Note
Applicable for any arithmetic progression, including consecutive natural, even, or odd numbers.
- Expression
Average = (First Term + Last Term) / 2
- Name
New Average after Adding an Item
- Note
When a single new observation is added to the group.
- Expression
New Average = ( (Old Average × Old Count) + New Item Value ) / (Old Count + 1)
- Name
New Average after Removing an Item
- Note
When a single observation is removed from the group.
- Expression
New Average = ( (Old Average × Old Count) - Removed Item Value ) / (Old Count - 1)
- Name
New Average after Replacing an Item
- Note
When one observation is replaced by another; the count remains unchanged.
- Expression
New Average = ( (Old Average × Old Count) - Replaced Item Value + New Item Value ) / Old Count
Prerequisites
Basic arithmetic operations (addition, subtraction, multiplication, division)
Understanding of fractions and decimals
Basic algebraic manipulation
Common mistakes
Incorrectly calculating the sum of observations.
Forgetting to adjust the 'number of observations' when items are added or removed.
Confusing simple average with weighted average in relevant scenarios.
Making calculation errors with large numbers or decimals.
Not understanding the impact of changes (addition, removal, replacement) on the average.
Keywords
Average
Mean
Arithmetic Mean
Weighted Average
Sum of Observations
Number of Observations
Consecutive Numbers
Practice preview
The average weight of 10 boys is 40 kg and the average weight of 5 girls is 30 kg. What is the average weight of all 15 students?…
medium
The average age of 20 students in a class is 15 years. If the age of the teacher is included, the average age increases by 1 year. What is the age of the teacher?…
medium
The average of 7 consecutive numbers is 20. What is the largest number?…
medium